Title:
A geometric take on Kostant's Convexity Theorem
Abstract: We characterize convex subsets of Rn
invariant under the linear action of a compact group G, by identifying
their images in the orbit space Rn/G by a purely metric
property. As a consequence, we obtain a version of Kostant's
celebrated Convexity Theorem (1973) whenever the orbit space
Rn/G is isometric to another orbit space
Rm/H. (In the classical case G acts by the adjoint
representation on its Lie algebra Rn, and H is the Weyl
group acting on a Cartan sub-algebra Rm). Being purely
metric, our results also hold when the group actions are replaced with
submetries.